Complex Analysis

This module introduces the theory of functions of a complex variable, statring with an introduction to complex numbers and ending with applications of the Residue Theorem and conformal transformations.

Learning Outcomes
1. represent a complex number as a point in the plane; calculate the modulus and argument of a complex number; switch between cartesian and polar forms; calculate the n-th roots of a complex number
2. decide where a function is diffrentiable (resp. analytic) using the Cauchy-Riemann equations
3. calculate the complex derivative of a function; decide whether a function is harmonic; calculate the harmonic conjugate of a harmonic function;
4. do various calculations involving exponentials and logarithms
5. parameterize a variety of paths in the plane
6. calculate the integral of a function along a given path
7. apply Cauchy's Theorem to compute integrals; apply Cauchy's Integral Formula to calculate various integrals;
8. calculate the Taylor series of a variety of elementary functions;
9. deduce the Laurent series of a range of functions;
10. apply the Residues Theorem to calculate various improper integrals.